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THE NORMAL DISTRIBUTION Lesson 1. Objectives To introduce the normal distribution The standard normal distribution Finding probabilities under the curve.

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Presentation on theme: "THE NORMAL DISTRIBUTION Lesson 1. Objectives To introduce the normal distribution The standard normal distribution Finding probabilities under the curve."— Presentation transcript:

1 THE NORMAL DISTRIBUTION Lesson 1

2 Objectives To introduce the normal distribution The standard normal distribution Finding probabilities under the curve

3 Normal Distributions Normal distributions are used to model continuous variables in many different situations. For example, a normal distribution could be used to model the height of students. We can transform our normal distribution into a standard normal distribution…

4 The standard normal variable The curve is designed so that the total area underneath the curve is 1 The curve fits within ±4 standard deviations from the mean

5 Find P (Z≤a) To do this we begin with a sketch of the normal distribution

6 P (Z≤a) To do this we begin with a sketch of the normal distribution. We then mark a line to represent Z=a a

7 P (Z≤a) To do this we begin with a sketch of the normal distribution. We then mark a line to represent Z=a P(Z≤a) is the area under the curve to the left of a. For continuous distributions there is no difference between P(Z≤a) and P(Z { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://slideplayer.com/4286177/14/images/slide_6.jpg", "name": "P (Z≤a) To do this we begin with a sketch of the normal distribution.", "description": "We then mark a line to represent Z=a P(Z≤a) is the area under the curve to the left of a. For continuous distributions there is no difference between P(Z≤a) and P(Z

8 Ex1 Find P (Z<1.55) To do this we begin with a sketch of the normal distribution

9 Ex 1 Find P (Z<1.55) To do this we begin with a sketch of the normal distribution. We then mark a line to represent Z=1.55 a

10 Ex 1 Find P (Z<1.55) To do this we begin with a sketch of the normal distribution. We then mark a line to represent Z=a P(Z<1.55) is the area under the curve to the left of a. We now use the table to look up this probability a

11 The Normal Distribution Table The table describes the positive half of the bell shaped curve…  (Z) is sometimes used as shorthand for P(Z { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://slideplayer.com/4286177/14/images/slide_10.jpg", "name": "The Normal Distribution Table The table describes the positive half of the bell shaped curve…  (Z) is sometimes used as shorthand for P(Z

12 The Normal Distribution Table

13 Ex 1 Find P (Z<1.55) To do this we begin with a sketch of the normal distribution. We then mark a line to represent Z=a P(Z<1.55) is the area under the curve to the left of a. We now use the table to look up this probability P(Z<1.55) = a

14 Ex 2 Find P(Z>1.74)

15 Note this result.. P(Z>a) = 1-P(Z1.74) = 1 – P(Z<1.74) = 1 – (0.9591) =

16 Ex 3 P(Z<-0.83) As our table only has values for the positive side of the distribution we must use symmetry…

17 Ex 3 P(Z<-0.83) We have reflected the curve in the vertical axis. P(Z<-0.83) = 1- P(0.83) = 1 – (0.7967) = This is a really useful technique P(Z a) = 1- P(Z { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://slideplayer.com/4286177/14/images/slide_16.jpg", "name": "Ex 3 P(Z<-0.83) We have reflected the curve in the vertical axis.", "description": "P(Z<-0.83) = 1- P(0.83) = 1 – (0.7967) = 0.2033 This is a really useful technique P(Z a) = 1- P(Z

18 This is a really useful result P(Z<-a) = 1 - P(Z { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://slideplayer.com/4286177/14/images/slide_17.jpg", "name": "This is a really useful result P(Z<-a) = 1 - P(Z

19 Ex 4 P(-1.24 { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://slideplayer.com/4286177/14/images/slide_18.jpg", "name": "Ex 4 P(-1.24

20 P(Z<2.16) =

21 Ex 4 P(-1.24 { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://slideplayer.com/4286177/14/images/slide_20.jpg", "name": "Ex 4 P(-1.24

22 Ex 4 P(-1.24 { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://slideplayer.com/4286177/14/images/slide_21.jpg", "name": "Ex 4 P(-1.24

23 Recommended work… Read through pages 177 and 178. Do Exercise 9A p179

24 Blank Normal Distribution Plot


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